Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}x+2y &= 1 \\ -7x+6y &= -5\end{align*}$
Begin by moving the $y$ -term in the second equation to the right side of the equation. $-7x = -6y-5$ Divide both sides by $-7$ to isolate $x$ $x = {\dfrac{6}{7}y + \dfrac{5}{7}}$ Substitute this expression for $x$ in the first equation. $({\dfrac{6}{7}y + \dfrac{5}{7}}) + 2y = 1$ $\dfrac{6}{7}y + \dfrac{5}{7} + 2y = 1$ Simplify by combining terms, then solve for $y$ $\dfrac{20}{7}y + \dfrac{5}{7} = 1$ $\dfrac{20}{7}y = \dfrac{2}{7}$ $y = \dfrac{1}{10}$ Substitute $\dfrac{1}{10}$ for $y$ in the top equation. $x+2( \dfrac{1}{10}) = 1$ $x+\dfrac{1}{5} = 1$ $x = \dfrac{4}{5}$ The solution is $\enspace x = \dfrac{4}{5}, \enspace y = \dfrac{1}{10}$.